Let \(u\in C^2(\overline{D})\) . Then, since \(p\in [2,\infty )\)  we get that \(u^{\langle p-1 \rangle } \in C^1(\overline{D})\)  and
\(\nabla u^{\langle p-1\rangle }(x) = (p-1)\nabla u(x) |u(x)|^{p-2},\quad x\in D.\) 
This puts us in a position to use Green's identity in the following way:
\(\int _D u^{\langle p-1 \rangle }(x)\Delta u(x)\, dx + (p-1)\int _D |\nabla u(x)|^2|u(x)|^{p-2}\, dx = -\int _{\partial D} u^{\langle p-1 \rangle }(w)\partial _{\vec{n}}^w u\, dw.\) 

Let \(v = P_D[u]\) , \(\phi = u - v\) , and note that \(\Delta \phi = \Delta u\)  and \(\phi = 0\)  (and so, \(u=v\) ) on \(\partial D\) . Furthermore,
\(\partial _{\vec{n}}^w u = \partial _{\vec{n}}^w v + \partial _{\vec{n}}^w\phi .\) 

Since \(u\)  is \(C^2(\overline{D})\)  we have \(\Delta \phi = \Delta u =f\in C(\overline{D})\)  and \(\phi (x) = -\frac{1}{2}\int _D G_D(x,y) f(y)\, dy\)  (see, e.g., Øksendal [1]} and [2]}). Therefore, by using an argument similar to the one in [3]} we get that
\(\partial _{\vec{n}}^w \phi = -\frac{1}{2}\lim \limits _{h\rightarrow 0^+}\int _D\frac{G_D(y,w+h\vec{n})}{h} f(y)\, dy = -\frac{1}{2}\int _D P_D(y,w)f(y)\, dy.\) 

Note that this means that both derivatives on the right-hand side of (REF ) exist. By Fubini's theorem,
\(\int _{\partial D} u^{\langle p-1 \rangle }(w)\partial _{\vec{n}}^w \phi \, dw &= -\frac{1}{2}\int _{\partial D} u^{\langle p-1 \rangle }(w)\int _D P_D(y,w) f(y)\, dy \, dw\nonumber \\&= -\frac{1}{2}\int _D f(y) \int _{\partial D}u^{\langle p-1 \rangle }(w) P_D(y,w)\, dw\, dy\nonumber \\&=-\frac{1}{2}\int _D \Delta u(y)P_D[u^{\langle p-1\rangle }](y) dy.\) 

By Grisvard [4]} we have \(\phi \in W^{2,2}(D)\) , and so \(v\in W^{2,2}(D)\)  as well. Since \(v\)  is smooth in \(D\) , this further yields \(v^{\langle p-1 \rangle }\in W^{1,2}(D)\) . By using Green's identity [4]} and the Douglas identity of Theorem REF  we find that
\(\int _{\partial D}u^{\langle p-1\rangle }(w)\partial _{\vec{n}}^w v\, dw = \int _{\partial D}v^{\langle p-1\rangle }(w)\partial _{\vec{n}}^w v\, dw &= -(p-1)\int _D|\nabla v(x)|^2|v(x)|^{p-2}\, dx\\&=-\frac{1}{p} \mathcal {H}_{\partial D}^p[u].\) 

Putting this together with (REF ) and (REF ) we get
\(-p\int _{\partial D}u^{\langle p-1\rangle }(w)\partial _{\vec{n}}^w u\, dw &= \mathcal {H}_{\partial D}^p[u] + \frac{p}{2}\int _D \Delta u(y)P_D[u^{\langle p-1\rangle }](y) dy\\&= \mathcal {E}_D^p[P_D[u]] + \frac{p}{2}\int _D \Delta u(y)P_D[u^{\langle p-1\rangle }](y) dy.\) 

By this and (REF ) we obtain the desired identities.
